Problem: Find the minimum value of
\[f(x) = x + \frac{1}{x} + \frac{1}{x + \frac{1}{x}}\]for $x > 0.$
First, consider the function
\[g(x) = x + \frac{1}{x}.\]If $1 \le x < y,$ then
\begin{align*}
g(y) - g(x) &= y + \frac{1}{y} - x - \frac{1}{x} \\
&= y - x + \frac{1}{y} - \frac{1}{x} \\
&= y - x + \frac{x - y}{xy} \\
&= (y - x) \left( 1 - \frac{1}{xy} \right) \\
&= \frac{(y - x)(xy - 1)}{xy} \\
&> 0.
\end{align*}Thus, $g(x)$ is increasing on the interval $[1,\infty).$

By AM-GM (and what we just proved above),
\[x + \frac{1}{x} \ge 2,\]so
\[g \left( x + \frac{1}{x} \right) \ge 2 + \frac{1}{2} = \frac{5}{2}.\]Equality occurs when $x = 1,$ to the minimum value of $f(x)$ for $x > 0$ is $\boxed{\frac{5}{2}}.$

In particular, we cannot use the following argument: By AM-GM,
\[x + \frac{1}{x} + \frac{1}{x + \frac{1}{x}} \ge 2 \sqrt{\left( x + \frac{1}{x} \right) \cdot \frac{1}{x + \frac{1}{x}}} = 2.\]However, we cannot conclude that the minimum is 2, because equality can occur only when $x + \frac{1}{x} = 1,$ and this is not possible.